Cryptography Reference
In-Depth Information
11.4 Information Theory of Cryptosystems
Where is the wisdom we have lost in knowledge?
Where is the knowledge we have lost in information?
T.S. Eliot
(1888-1965), Anglo-American poet, critic, and dramatist
— from
The Rock
(1934)
When we defined conditional entropyin Equation (11.3) on page 431, we
looked at a cryptological context. It is this interpretation upon which we now
concentrate. In fact, the quantitydefined in that context for Equation (11.3) is
called
key equivocation
.
Key Equivocation
The entropy of cryptosystems is a key feature upon which we will focus
herein. A cryptosystem may be defined by parameters that include the keyspace
K
(as well as encryption and
decryption transformations), and certain probability distributions given as fol-
lows. Each plaintext unit,
m
, the message space
M
, the ciphertext space
C
∈
M
, has a certain probabilityof occurring, and
the choice of key
k
∈
K
is assumed to be independent of the choice of
m
, with
probabilityof a given
k
∈
K
also having a probabilitydistribution from which
it follows that
H
(
K
,
M
)=
H
(
K
)+
H
(
M
)
(see part 1 of The Role of Independence on page 432). Also, the possible
c
∈
C
have a probabilitydistribution that depends on the probabilitydistributions for
M
K
and
. Given this setup, the keyequivocation satisfies
K
|
C
K
M
−
C
H
(
)=
H
(
)+
H
(
)
H
(
)
,
(11.8)
which is a measure of how much information about the keyis revealed bythe
ciphertext.
Example 11.7
Let
M
=
{
s
1
,s
2
,s
3
,s
4
}
with probabilities,
p
s
1
=0
.
1
,
s
2
=0
.
2
,
s
3
=0
.
3
,
and
p
s
4
=0
.
4;
K
=
{
k
1
,k
2
,k
3
}
with probabilities,
p
k
1
=0
.
3
,
k
2
=0
.
3
,
and
p
k
3
=0
.
4;
and
.
If
E
k
is the enciphering transformation for a given
k
C
=
{
c
1
,c
2
,c
3
,c
4
}
∈
K
, and
E
k
1
(
s
1
)=
c
1
;
E
k
1
(
s
2
)=
c
2
;
E
k
1
(
s
3
)=
c
3
;
E
k
1
(
s
4
)=
c
4
E
k
2
(
s
1
)=
c
2
;
E
k
2
(
s
2
)=
c
3
;
E
k
2
(
s
3
)=
c
4
;
E
k
2
(
s
4
)=
c
1
E
k
3
(
s
1
)=
c
3
;
E
k
3
(
s
2
)=
c
4
;
E
k
3
(
s
3
)=
c
1
;
E
k
3
(
s
4
)=
c
2
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